On Surface Approximation Using Developable Surfaces

Chen, H.-Y.; Lee, In-Kwon; Leopoldseder, Stefan; Pottmann, Helmut; Randrup, Thomas B.; Wallner, Johannes

doi:10.1006/gmip.1999.0487

Cited by 64 publications

(34 citation statements)

References 14 publications

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“…Therefore, relating to our quasidevelopable mesh surface interpolation problem, our objective is to find one "most" developable mesh surface to satisfy the given boundary constraints. This is different from the problem of fitting a point cloud with a single "100 percent" developable torsal surface (cf., [8] and [9]), wherein the objective is to minimize the interpolation error.…”

Section: Background and Previous Workmentioning

confidence: 99%

Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

Tang

Chen

2009

IEEE Trans. Visual. Comput. Graphics

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Abstract-We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and overdetermined linear system that is subsequently solved by a robust least squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental tests have shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.

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Section: Background and Previous Workmentioning

confidence: 99%

Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

Tang

Chen

2009

IEEE Trans. Visual. Comput. Graphics

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“…Research related to Computer Aided Geometric Design, in particular those concerning the design and approximation of developable surfaces, can be found in [18][19][20][21][22][23][24][25][26][27]. Most of them are in terms of NURBS or its special case -B-spline or Bézier surfaces [18][19][20][21][22][23][24]. Aumann [18] proposed the condition under which a developable Bézier surface can be constructed with two boundary curves.…”

Section: Related Workmentioning

confidence: 99%

“…In their method, after one boundary curve is freely specified, five more degrees of free are available for a second boundary curve of the same degree. In the work of [22][23][24], the approximation methods are used to design developable B-Spline surfaces based on projective geometry. Other approaches are based on alternative perspective: Redont [25] constructs developable surfaces by specifying tangent planes along a geodesic of a surface, Randrup [26] approximates a given surface by cylinders in its Gaussian image, and Park et al [27] design developable surfaces by the methods from optimal control theory.…”

Section: Related Workmentioning

confidence: 99%

Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches

Wang

Tang

2004

Vis Comput

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Surface developability is required in a variety of applications in product design, such as clothing, ship hulls, automobile parts, etc. However, most current geometric modeling systems using polygonal surfaces ignore this important intrinsic geometric property. This paper investigates the problem of how to minimally deform a polygonal surface to attain developability, or the so called developability-by-deformation problem. In our study, this problem is first formulated as a global constrained optimization problem, and a penalty function based numerical solution is proposed for solving this global optimization problem. Next, as an alternative to the global optimization approach which usually requires lengthy computing time, we present an iterative solution based on a local optimization criterion which achieves near real-time computing speed. Both approaches preserve the topology and continuity of the original polygonal surface in the case when more than one individual polygonal patches comprise the surface. Experimental examples are provided to demonstrate the functionality of the proposed two approaches as well as their comparison in terms of computing cost, effectiveness of attaining developability, dimensional difference between the surfaces before and after the optimization, and other important aspects.

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“…In some cases curve reconstruction plays an important role in the surface reconstruction problem [3,20,21,22]. In this paper, we focus on the reconstruction of a curve from an unorganized point cloud having no ordering of the point elements.…”

Section: Introductionmentioning

confidence: 99%

“…An attractive feature of our solution is that the algorithm can be extended to any dimension. The motivation of this work comes from recent research attempting to reconstruct surfaces of revolution, helical surfaces, spiral surfaces, profile surfaces and developable surfaces from a set of points [3,20,21,22]. In that research, principal motion parameters, such as motion axis and pitch, are computed using line geometry.…”

Section: Introductionmentioning

confidence: 99%

Curve reconstruction from unorganized points

Lee

2000

Computer Aided Geometric Design

245

132

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We present an algorithm to approximate a set of unorganized points with a simple curve without self-intersections. The moving least-squares method has a good ability to reduce a point cloud to a thin curve-like shape which is a near-best approximation of the point set. In this paper, an improved moving least-squares technique is suggested using Euclidean minimum spanning tree, region expansion and refining iteration. After thinning a given point cloud using the improved moving least-squares technique we can easily reconstruct a smooth curve. As an application, a pipe surface reconstruction algorithm is presented.

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On Surface Approximation Using Developable Surfaces

Cited by 64 publications

References 14 publications

Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches

Curve reconstruction from unorganized points

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